Large scale phenomena in models of statistical mechanics

该项目解决了概率论和统计力学边缘的各种数学问题。它们的共同特点是公式相对简单，潜在现象有趣，与物理学科有很深的联系。第一轮问题是关于随机电导模型的。其中包括无序介质中随机游走的标度极限的推导，随机拉普拉斯算子的谱性质的研究，有效电导率波动的分析，热核衰减的详细控制等。第二个要研究的领域是梯度场。这些问题在物理和应用科学中普遍存在，但由于无法克服潜在相互作用的非凸性，它们的数学理解目前处于停滞状态。我们提供了具体的方法和途径，如何在技术上克服这个障碍。第三类问题涉及无序系统领域。感兴趣的具体主题是随机场自旋模型和由抛物线安德森模型控制的动力学。本文概述了精确的数学方法，这些方法可以显著提高对这些问题的数学理解。列出的许多问题的设计目的是为对概率论和数学物理感兴趣的研究生和博士后提供培训和纳入研究。硬科学在精确描述、建模甚至预测复杂自然现象方面的巨大成功，在很大程度上源于它们以严谨的数学为基础。在固态物理和材料科学的背景下，最常用的数学方法是概率和/或统计方法。这并不奇怪，因为这些学科的设计正是为了处理涉及大量个体成分的系统。本项目研究大系统概率论中的三种特定类型的问题，这些问题的根源是材料物理学。我们提出的最相关的一般性问题是，材料的结构细节及其各种固有的不规则性如何准确地在其宏观性质中表达出来。通常，我们寻求一种包络性的原则，或一种“物理定律”，以定量或定性的形式从细节中提取出本质特征。希望发展这种系统的理论基础将最终导致工程应用的重大进展。这对数学本身也有直接的影响：对复杂现象的分析不可避免地涉及到许多独立的数学分支学科，因此将导致它们之间富有成效的思想交流。该项目自然为研究生和博士后提供了一个进入美国顶尖研究型大学研究环境的机会。

The project addresses a variety of mathematical problems at the borderline of probability theory and statistical mechanics. The common feature of these is a relatively simple formulation, intriguing underlying phenomena and deep connections to physical disciplines. The first round of problems pertains to the Random Conductance Model. These include derivation of scaling limits of random walks in disordered media, study of spectral properties of random Laplacians, analysis of fluctuations of effective conductivity, detailed control of heat-kernel decay, etc. The second area proposed to investigate is that of Gradient Fields. These problems are ubiquitous in physical and applied sciences but their mathematical understanding is currently stalled because of the inability to overcome non-convexity of the underlying interaction. We offer specific methods and approaches how this obstacle may technically be overcome. The third class of problems addresses the area of Disordered Systems. The specific subjects of interest are random-field spin models and the dynamics governed by the parabolic Anderson model. Precise mathematical approaches are outlined that could lead to significant improvements in the mathematical understanding of these problems. A good many of listed problems are devised with the intention to provide training, and inclusion in research, of graduate students and postdocs who have interest in Probability Theory and Mathematical Physics.The tremendous success of hard sciences in accurate description, modeling and even forecasting complex natural phenomena derives, in large part, from their foundation in rigorous mathematics. In the context of solid-state physics and material sciences, the mathematical methods most commonly used are those of probability and/or statistics. This comes as no surprise as these disciplines have been designed precisely to deal with systems involving large numbers of individual constituents. The present project studies three specific classes of problems in probability theory of large systems whose origin is rooted in physics of materials. The most pertinent general question we ask is how the structural details of materials, and their various inherent irregularities, exactly express themselves in their macroscopic properties. As a rule, we seek an enveloping principle, or a "physical law", that extracts, in quantitative or qualitative form, the essential features from the specifics. It is hoped that developing the theoretical foundations of such systems will eventually lead to significant advances in engineering applications. There is also an immediate impact for mathematics itself: analysis of complex phenomena inevitably involves a number of separate sub-disciplines of mathematics and will thus lead to fruitful exchange of ideas among them. The project naturally offers an opportunity to include graduate students and postdocs into the research environment at a top research US university.